Simplify ( square root of 14(cos(85)+isin(85)))^6

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Trigonometry

${(\sqrt{14} (\cos (85) + i \sin (85)))}^{6}$

Simplify terms.

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Simplify each term.

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Evaluate $\cos (85)$ .

${(\sqrt{14} (0.08715574 + i \sin (85)))}^{6}$

Evaluate $\sin (85)$ .

${(\sqrt{14} (0.08715574 + i \cdot 0.99619469))}^{6}$

Move $0.99619469$ to the left of $i$ .

${(\sqrt{14} (0.08715574 + 0.99619469 i))}^{6}$

Simplify terms.

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Apply the distributive property.

${(\sqrt{14} \cdot 0.08715574 + \sqrt{14} (0.99619469 i))}^{6}$

Multiply $\sqrt{14}$ by $0.08715574$ .

${(0.32610692 + \sqrt{14} (0.99619469 i))}^{6}$

Multiply $0.99619469$ by $\sqrt{14}$ .

${(0.32610692 + 3.72741925 i)}^{6}$

Use the Binomial Theorem.

${0.32610692}^{6} + 6 \cdot {0.32610692}^{5} (3.72741925 i) + 15 \cdot {0.32610692}^{4} {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Simplify terms.

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Simplify each term.

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Raise $0.32610692$ to the power of $6$ .

$0.0012027 + 6 \cdot {0.32610692}^{5} (3.72741925 i) + 15 \cdot {0.32610692}^{4} {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $0.32610692$ to the power of $5$ .

$0.0012027 + 6 \cdot 0.00368807 (3.72741925 i) + 15 \cdot {0.32610692}^{4} {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $6$ by $0.00368807$ .

$0.0012027 + 0.02212846 (3.72741925 i) + 15 \cdot {0.32610692}^{4} {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $3.72741925$ by $0.02212846$ .

$0.0012027 + 0.08248208 i + 15 \cdot {0.32610692}^{4} {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $0.32610692$ to the power of $4$ .

$0.0012027 + 0.08248208 i + 15 \cdot 0.01130941 {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $15$ by $0.01130941$ .

$0.0012027 + 0.08248208 i + 0.16964121 {(3.72741925 i)}^{2} + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Apply the product rule to $3.72741925 i$ .

$0.0012027 + 0.08248208 i + 0.16964121 ({3.72741925}^{2} i^{2}) + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $3.72741925$ to the power of $2$ .

$0.0012027 + 0.08248208 i + 0.16964121 (13.89365427 i^{2}) + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Rewrite $i^{2}$ as $- 1$ .

$0.0012027 + 0.08248208 i + 0.16964121 (13.89365427 \cdot - 1) + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $13.89365427$ by $- 1$ .

$0.0012027 + 0.08248208 i + 0.16964121 \cdot - 13.89365427 + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $0.16964121$ by $- 13.89365427$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 20 \cdot {0.32610692}^{3} {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $0.32610692$ to the power of $3$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 20 \cdot 0.03468007 {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $20$ by $0.03468007$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 {(3.72741925 i)}^{3} + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Apply the product rule to $3.72741925 i$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 ({3.72741925}^{3} i^{3}) + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $3.72741925$ to the power of $3$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 (51.78747439 i^{3}) + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Factor out $i^{2}$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 (51.78747439 (i^{2} \cdot i)) + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Rewrite $i^{2}$ as $- 1$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 (51.78747439 (- 1 \cdot i)) + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Rewrite $- 1 i$ as $- i$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 (51.78747439 (- i)) + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $- 1$ by $51.78747439$ .

$0.0012027 + 0.08248208 i - 2.35693633 + 0.69360158 (- 51.78747439 i) + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $- 51.78747439$ by $0.69360158$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 15 \cdot {0.32610692}^{2} {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $0.32610692$ to the power of $2$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 15 \cdot 0.10634572 {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $15$ by $0.10634572$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 {(3.72741925 i)}^{4} + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Apply the product rule to $3.72741925 i$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 ({3.72741925}^{4} i^{4}) + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $3.72741925$ to the power of $4$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 (193.033629 i^{4}) + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Rewrite $i^{4}$ as $1$ .

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Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 (193.033629 {(i^{2})}^{2}) + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Rewrite $i^{2}$ as $- 1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 (193.033629 {(- 1)}^{2}) + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Raise $- 1$ to the power of $2$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 (193.033629 \cdot 1) + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $193.033629$ by $1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 1.59518593 \cdot 193.033629 + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $1.59518593$ by $193.033629$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 6 \cdot 0.32610692 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Multiply $6$ by $0.32610692$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 {(3.72741925 i)}^{5} + {(3.72741925 i)}^{6}$

Apply the product rule to $3.72741925 i$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 ({3.72741925}^{5} i^{5}) + {(3.72741925 i)}^{6}$

Raise $3.72741925$ to the power of $5$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 (719.51726479 i^{5}) + {(3.72741925 i)}^{6}$

Factor out $i^{4}$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 (719.51726479 (i^{4} i)) + {(3.72741925 i)}^{6}$

Rewrite $i^{4}$ as $1$ .

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Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 (719.51726479 ({(i^{2})}^{2} i)) + {(3.72741925 i)}^{6}$

Rewrite $i^{2}$ as $- 1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 (719.51726479 ({(- 1)}^{2} i)) + {(3.72741925 i)}^{6}$

Raise $- 1$ to the power of $2$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 (719.51726479 (1 i)) + {(3.72741925 i)}^{6}$

Multiply $i$ by $1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1.95664157 (719.51726479 i) + {(3.72741925 i)}^{6}$

Multiply $719.51726479$ by $1.95664157$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + {(3.72741925 i)}^{6}$

Apply the product rule to $3.72741925 i$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + {3.72741925}^{6} i^{6}$

Raise $3.72741925$ to the power of $6$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 i^{6}$

Factor out $i^{4}$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 (i^{4} i^{2})$

Rewrite $i^{4}$ as $1$ .

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Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 ({(i^{2})}^{2} i^{2})$

Rewrite $i^{2}$ as $- 1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 ({(- 1)}^{2} i^{2})$

Raise $- 1$ to the power of $2$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 (1 i^{2})$

Multiply $i^{2}$ by $1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 i^{2}$

Rewrite $i^{2}$ as $- 1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i + 2681.94250408 \cdot - 1$

Multiply $2681.94250408$ by $- 1$ .

$0.0012027 + 0.08248208 i - 2.35693633 - 35.91987409 i + 307.92452972 + 1407.83739201 i - 2681.94250408$

Simplify by adding terms.

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Subtract $2.35693633$ from $0.0012027$ .

$- 2.35573362 + 0.08248208 i - 35.91987409 i + 307.92452972 + 1407.83739201 i - 2681.94250408$

Simplify by adding and subtracting.

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Add $- 2.35573362$ and $307.92452972$ .

$305.56879609 + 0.08248208 i - 35.91987409 i + 1407.83739201 i - 2681.94250408$

Subtract $2681.94250408$ from $305.56879609$ .

$- 2376.37370798 + 0.08248208 i - 35.91987409 i + 1407.83739201 i$

Subtract $35.91987409 i$ from $0.08248208 i$ .

$- 2376.37370798 - 35.83739201 i + 1407.83739201 i$

Add $- 35.83739201 i$ and $1407.83739201 i$ .

$- 2376.37370798 + 1371.99999999 i$

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